\(\left|x-4\right|\sqrt{3x+1}\ge0\)
\(\rightarrow\left[{}\begin{matrix}\left(x-4\right)\sqrt{3x+1}\ge0\\\left(x-4\right)\sqrt{3x+1}\le0\end{matrix}\right.\)
\(\rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-4\ge0\\\sqrt{3x+1}\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}x-4\le0\\\sqrt{3x+1}\le0\end{matrix}\right.\\\left\{{}\begin{matrix}\left(x-4\right)\ge0\\\sqrt{3x+1}\le0\end{matrix}\right.\\\left\{{}\begin{matrix}x-4\le0\\\sqrt{3x+1}\ge0\end{matrix}\right.\end{matrix}\right.\)
\(\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge4\\x\ge\frac{-1}{3}\end{matrix}\right.\\\left\{{}\begin{matrix}x\le4\\x\le\frac{-1}{3}\end{matrix}\right.\\\left\{{}\begin{matrix}x\ge4\\x=\frac{-1}{3}\end{matrix}\right.\\\left\{{}\begin{matrix}x\le4\\x\ge\frac{-1}{3}\end{matrix}\right.\end{matrix}\right.\rightarrow\left[{}\begin{matrix}x\ge4\\x=\frac{-1}{3}\\\frac{-1}{3}\le x\le4\end{matrix}\right.\)
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